Richard Barsotti

July 10th. 2013

MECHANICS 8.0: VISCOSITY

Viscosity

The drag acting on a motorcycle and rider is produced by two different mechanisms: pressure changes and skin friction. The latter is caused by the layer of air very close to moving surfaces--the so-called boundary layer. The boundary layer and its behavior is controlled by its viscosity. Outside the boundary layer, the flow is inviscid. Although this free-stream flow is frictionless it can affect the boundary layer. Which, in turn, affects the pressure distribution around the bike, particularly at the rear section. In the case of a moving motorcycle, more drag is created by reduced pressure behind it--the base drag--than by increased pressure in front of it. Although counterintuitive, this fact is important because it is the pressure distribution that is largely responsible for the drag forces that are generated. The point at which the total aerodynamic force can be considered to act is called the center of pressure which is most unlikely to coincide exactly with the motorcycle’s center of gravity. Drag is always present but a great deal of aerodynamic loading depends on an object’s shape and the direction of airflow. Turning effects occur when the center of pressure is at odds with the center of gravity. Conservation of Energy is the fundamental principle behind pressure changes. Air in motion has energy due to its mass and velocity (kinetic energy), its atmospheric pressure, and its position in the gravitational field (potential energy). For the most part, gravitational potential energy can be ignored. Therefore, an approximation of the total energy present for air flow is given by:

Total Energy = Pressure Energy + Kinetic Energy.

Since the total energy is constant, if the kinetic energy of air increases then the pressure energy must decrease or conversely, if the air’s kinetic energy decreases its pressure energy increases. This form of Bernoulli’s Theorem neglects the very important fact that air has viscosity (cold air being less viscous). In the absence viscosity, a cylinder moving through the air would demonstrate a symmetrical pressure distribution that would not cause drag--viscosity ensures that all shapes moving relative to a fluid produce drag.

Viscous Flow: Two familiar forms of bulk matter are solids and fluids, the latter being further subdivided into liquids and gases. If one includes plasma (gases with significant charged-particle content) in the fluids category, then more than 99.99% of bulk matter in the universe is in the fluid state. Intermolecular forces dominate in the closely spaced arrangement of a solid. They have a smaller, but important role in a liquid and are very minute in a gas. As a consequence, liquids and gases will flow (generally deform when subjected to a shearing force no matter now small), such behavior can, in fact, be taken as a working definition of a fluid.
Although fluid static and related dynamic properties are ultimately explained at the molecular level, the largest segment of fluid physics deals with bulk behavior, that is, on a macroscopic scale. This classical view involves enough molecules so that “local” quantum fluctuations will be smoothed-out with negligible effect on the measured average values of physical quantities.
The study of fluids in motion, arguably, began when Leonardo da Vinci (1452-1519) observed and drew complex flows over objects in streams. A quantitative physical and mathematical understanding of fluid flow began when Isaac Newton (1642-1727) devoted Book II of his Principia Mathematica (1687) exclusively to the examination of fluids. However, it was Leonhard Euler (1707-1783) who was most instrumental in mathematically conceptualizing fluid dynamics. He described flow in terms of spatially varying three-dimensional pressure and velocity fields and modeled flow as a continuous collection of infinitesimally small fluid elements. While Euler’s coupled nonlinear partial differential equations were a theoretical breakthrough in fluid dynamics, finding general solutions for them proved very difficult. More importantly, Euler did not account for the effect of friction acting on the motion of the fluid elements--that is, he ignored viscosity; which Newton described as “...a lack of slipperiness between adjacent layers of fluid.”
When a solid experiences forces that tend to twist it, it initially distorts under the shear stress, builds up a resistance to further distortion and soon reaches equilibrium in some new shape.
By contrast, a fluid bounded by two parallel planes in relative parallel motion will undergo a shearing strain in response to the applied stress. Propagation of the stress is possible because any real fluid (as opposed to an idealized one) adheres to the planes and exhibits
internal friction as well. Such behavior is termed viscosity and stems from the transfer of molecular momentum among flow patterns with different fluid velocities. A large variety of fluids will strain at a rate which is proportional to the applied stress--these are called Newtonian fluids: water, light oil, and air are typical.
For example, imagine a fixed flat surface coated with a layer of liquid of depth on top
of which is a plate of area . When a constant force is applied to the plate, it monentarily accelerates but soon moves with a constant terminal speed along the x-axis at which point the applied force is exactly opposed by a viscous counter force. If the applied force is then doubled, the terminal speed of the plate is found to double as well; indeed, is proportional to force . An increase in depth of the liquid results in a proportional increase in the speed of the plate. For any given constant force, is proportional to . Not surprisingly, with the other parameters kept constant, and increase in area A results in a decrease in terminal speed; hence, is inversely proportional to . Thus:

along the x-axis is proportional to or is proportional to .

Turning the second proportionality into and equality requires a constant of proportionality.

Hence, Equation 1: and knowing that is stress and here, in particular, it is shear stress we can obtain an equation for viscosity of Newtonian fluids:

Equation 2: . Where symbolizes shear stress.

The term is the velocity gradient, and it arises because the fluid speed changes with depth as it is dragged along by the moving plate. The liquid in contact with the plate adheres to it and so moves at terminal speed , where as the liquid in contact with the fixed surface is at rest, i.e., = 0. Newton proposed that the fluid behaves as if it consisted of very thin adjacent sheets or laminae that move with respect to each other. Such a model improves as the number of laminae increases and their respective thickness decreases,
hence, Newton’s Law of Laminar Viscosity becomes:

Equation 3: .

Note that fluid pressure does not appear in either of the above equations, showing how completely different fluid friction is compared to friction involving solids. Also the relationship appears to be independent of fluid density, which implies an independence from macroscopic inertial fluid features as well. For certain flow regimes this is not true--see addendum to this article.
Viscosity is measured in Newton-second/meter squared and in poise (after J.L.M. Poiseuille--pronounced pwah zoy), 1 poise = 1 dyne-second/centimeter squared or more commonly, in centipoise.
As the following table revels, viscosity is a function of temperature; increasing for gases and decreasing for liquids.

Table 1: Viscosity of Some Common Fluids

Liquids Temperature (Celsius) Viscosity (centipoise)
Acetone 20 0.3200
Benzene 20 0.6520
Blood plasma 37 1.300
Blood whole 37 2.080
Ethyl alcohol 20 1.200
25 1.095
Gasoline 20 0.600
Glycerine 20 1.490
Liquid Sodium 250 0.3810
Mercury 25 1.530
Oil (Castor) 25 0.6500
Oil (heavy) 20 0.6600
Oil (light) 20 0.1100
Water 0 1.798
20 1.002
100 0.2818
Gases
Air 0 0.0171
20 0.0180
40 0.0190
Argon 20 0.0022
Ammonia 20 0.0098
Carbon Dioxide 20 0.0148
Helium 20 0.0140
Methane 20 0.0190
Oxygen 20 0.0200
Steam 100 0.0130

In the mid 1830s Poiseuille conducted a series of experiments to determine the way viscous fluids propagated through pipes. His result can be understood by imagining a fluid in a pipe to consist of adjacent concentric cylinders that move at diminishing speeds from the center. This is essentially a circular symmetry version of the experiment described above, where now the stationary surface (at which = 0) is the inside face of the pipe. The idea being, to derive an equation for v as a function of radius r outward from the central x-axis and then determine the discharge rate .
Think of a pipe of overall radius R filled with concentric cylinders of fluid, each of radius r and an outer surface area . If the fluid is moving with a constant speed along the x-axis then, from Eqs.1& 3 above, the viscous drag force on each concentric cylinder is in the opposite direction or,

Equation 4: and by substitution,

Equation 5: .

Assuming, in this basic situation, there is no acceleration, that negative drag force must be balanced by the force on each cylinder due to the pressure difference acting on its ends.
That pressure difference at the input and output surfaces and the force it generates equals. Therefore substituting in Eq. 5 and solving for dv,
one obtains:

and by integrating both sides

Equation 6: .

Notice, the speed profile is parabolic; the maximum occurring along the centerline where = 0.

The volume flux (in cubic meter/ second) passing through a differential concentric cylinder is given by: , where is volume. Therefore,

integrating over all elements
between and

or

Equation 7:
Reasonably enough, the flow rate increases with both the pressure gradient , and the radius and decreases with viscosity. The fourth power dependence on the radius is important. It arises because a narrow diameter pipe will have a much larger percentage of the fluid in direct contact with its walls than a wider pipe, and that contact inhibits flow: the

the volume of the differential fluid cylinder is while its surface area is . The former increases more rapidly as increases than does the latter. All other factors the same, a fluid in a small-diameter pipe will flow more slowly than the same fluid in a large-diameter pipe.

The relative importance of viscosity in a particular flow situation stems from the magnitude of the viscous terms as compared to other terms in the equation. Such comparisons in fluid dynamics leads to a non dimensional term called the Reynolds Number (after Osborne Reynolds, 1842-1912), which is (apart from a factor of 2) simply a ratio the dynamic pressure

to viscous stress , being fluid density, the fluid velocity, and a typical length of the fluid under study. Thus,

.

The Reynolds Number is the most important scaling parameter in aerodynamics and a controlling guide for overall behavior and analysis of a wide variety of fluid flows. For example, consider the steady motion of a sphere immersed in a fluid. At low Reynolds Numbers (1 or less), the inertial transfer of momentum can be ignored when compared to the surface shear or fluid “friction”. This behavior is known as Stokes flow. The drag force on a sphere of radius r moving with velocity relative to the fluid is, for Stokes flow:

.

If the Reynolds Number ranges around , the drag force on the sphere is given by:

.

Where is a constant somewhat less than but close to unity and rho is the fluid density.
Note the differences in the two Stokes drag equations the first equation depends linearly on viscosity and velocity , but not on density. That is, not on the inertial feature of flow. Conversely, the higher Reynolds Number equation does show a dependence on fluid density and a quadratic dependence on velocity, but no dependence on viscosity.

For the drag force is complicated by the transition from laminar to turbulent flow near the spherical surface and to changes in the location of flow separation from its surface.

Low-Reynolds-Number flows, e.g., glycerin flowing through a pipe, show viscous effects everywhere in the flow field. However, at high Reynolds Numbers( or greater), predominant regions away from the boundaries can be treated on an inviscid basis, while near the boundaries, relatively thin layers exist in which both viscous and inertial terms must be taken into account. These so-called boundary layers were first introduced by Ludwig Prandtl in 1904, and by now have been studied in great detail.

Consider an airfoil (a wing) in motion. The fluid exerts a net aerodynamic force on the airfoil originating from two sources: the fluid pressure and the shear-stress that results from friction between the surface and the flow. Fluid pressure acts normal (perpendicular) to the surface; while the shear-stress acts tangentially.
In the boundary layer (which is very, very thin compared to the size of the body under study) adjacent to the surface, the effects of friction are dominant. The flow velocity changes as a function of the normal distance , from zero at the surface to the full inviscid

flow value at the boundary’s outer edge. The velocity progression from zero to some frictionless value is exponential. Outside the boundary layer, the flow is essentially inviscid.

Furthermore, the boundary layer can separate from the top surface of an airfoil if the “angle of attack” is greater than the so-called “stall angle”. At the leading edge, the upper region that trails downstream from the separation point is the remnant of the boundary layer that originally formed on the top of the airfoil. The lower region of separation that extends downstream from the trailing edge of the airfoil is the remnant of the boundary layer over the bottom surface. When separated, these two regions are called shear layers, and they form the upper and lower extremes of the separated flow. Cohabiting the separated flow region are low energy flow forms in the wake behind the airfoil, and for the most part, dead air.

The pressure distribution over the surface of an airfoil is radically changed once the flow separates. The altered distribution creates a pressure drag due to the flow separation, that is, a large unbalanced force that acts in the direction of the free-stream flow--the drag direction. When the flow separation is extensive (when the separated flow region is large), the pressure drag is usually much greater than the skin friction.

In summation, aerodynamic flow over a body can be divided into two regions: a thin boundary layer near the surface, where friction dominates, and an inviscid flow external to the boundary layer where friction is negligible. The inviscid flow strongly affects boundary- layer properties; indeed, the external flow creates the boundary conditions at the outer edge of the boundary layer and dictates the velocity profile within the layer. On the other hand, the boundary layer is too thin to have any real effect on the external inviscid flow. the exception to the no effect rule is if the flow separates, then the inviscid flow is greatly modified by the presence of the separated region.
After Prandtl’s 1904 paper and subsequent work (by himself and graduate students) it became clear that, in most cases, viscosity only played a role in the thin layer immediately adjacent to the surface of a body immersed in a flowing fluid. That was a major breakthrough in understanding viscous flow and gave the fluid dynamist a means to calculate shin-friction drag. Before Prandtl there was no understanding of the physical mechanism that caused flow separation from a surface. His brilliant insight clarified the physics of separated flow and fluid dynamics entered the modern era just in time for powered flight.

Addendum: Fluid density is basic to the analysis of buoyancy and other static fluid phenomena, and contributes to the inertial behavior of fluids in dynamic situations. In fact, drag forces on a solid body are proportional to fluid density in certain flow regimes. In flows of gases with heat exchange or a velocities comparable to or larger than the speed of sound in the fluid, the density changes from constant to variable. This is important because such density variation or “compressibility” couples flow dynamics with flow energitics, resulting in behaviors which are quite different from incompressible, isothermal flows.
Kinematic viscosity (the ratio of the viscosity coefficient to fluid density) determines the diffuse spread of fluid straining behavior originating at a local disturbance in an otherwise uniform flow. Such transport is of the same nature as that of heat conduction or of mass diffusion in an non homogeneous fluid mixture. The latter two properties play
an important role, respectively, in fluid heat transfer and in chemically reactive fluid flows involving mixing.

Ref. The Racing Motorcycle Volume 1 John Bradley
Ref. Physics Today December, 2005 Ludwig Prandtl's Boundary Layer John D. Anderson Jr. 